The Decomposition of Figures Into Smaller Parts – Boltyanskii and Gohberg

In this post, we will see the book The Decomposition of Figures Into Smaller Parts by V. G. Boltyanskii and I. T. Gohberg. The book is part of the Popular Lectures in Mathematics Series.

About the book:

This book is devoted to some interrelated problems of a new, rapidly developing branch of mathematics called combinatorial geometry. Common to all the problems examined here is the notion of “ cutting” a geometric figure into several “smaller pieces.” There are several different criteria for what constitutes a “ smaller piece” ; hence this book necessarily treats several different problems. All the theorems proved here are very recent; the oldest of them was proved by the Polish mathematician Karol Borsuk about forty years ago. This theorem of Borsuk is the core around which all of the subsequent exposition unfolds. The most recent theorem is barely a year old.

The topics treated in this book are well within the grasp of bright and interested high school students. At the same time, the book intro­ duces the reader to a number of the unsolved problems of geometry.

This family of problems is the subject of another book by the same authors. Theorems and Problems in Combinatorial Geometry (Nauka, 1965). That book, however, deals chiefly with problems of three- dimensional and higher-dimensional spaces. The present book concerns itself only with problems of plane geometry, and can thus be used by high school mathematics clubs. Theorems and Problems in Combinatorial Geometry will be useful, however, to readers interested in continuing further.

The remarks at the end of the book are intended for the more advanced reader.

The book is part of the series Popular Lectures in Mathematics.

PDF | OCR | Bookmarked | 2.7 MB | 78 pages

All credits to the original uploader.

The Internet Archive link.



Preface vi

1. Division of Figures into Pieces of Smaller Diameter 1

1.1. The Diameter of a Figure 1
1.2. Formulation of the Problem 3
1.3. Borsuk’s Theorem 4
1.4. Convex Figures 7
1.5. Figures of Constant Width 12
1.6. Embedding in a Figure of Constant Width 14
1.7. For Which Figures is a{F) = 3? 19

2. Division of Figures in the Minkowski Plane 26

2.1. A Graphic Example 26
2.2. The Minkowski Plane 28
2.3. Borsuk’s Problem in Minkowski Planes 34

3. The Covering of Convex Figures by Reduced Copies 40

3.1. Formulation of the Problem 40
3.2. Another Formulation of the Problem 41
3.3. Solution of the Covering Problem 42
3.4. Proof of Theorem 2.2 52

4. The Problem of Illumination 55

4.1. Formulation of the Problem 55
4.2. Solution of the Problem of Illumination 57
4.3. The Equivalence of the Last Two Problems 58
4.4. Division and Illumination of Unbounded Convex Figures 63

Remarks 66



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3 Responses to The Decomposition of Figures Into Smaller Parts – Boltyanskii and Gohberg

  1. Very good! Thanks Mitr!
    Hei guys! Does anyone have this book?
    “Radio Transmitter Design” edited by Vagan V. Shakhgildyan
    Translated from the Russian by Boris Kuznetsov, MIR PUBLISHERS, MOSCOW.
    1987 Printing of the Revised 1984 Russian Edition, Hard Bound, 488 Pages.

  2. desperadomar says:

    After a loooooooooong time!!!!!!!!!!111

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